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Learning When to Act: Interval-Aware Reinforcement Learning with Predictive Temporal Structure

Davide Di Gioia

Year: 2026Venue: arXiv preprintArea: cs.LGType: PreprintEmbeddings: 70

Intelligence

Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 96%

Last extracted: 3/26/2026, 1:34:35 AM

Summary

The paper introduces Adaptive Temporal Control via Predictive Geometry (ATCPG), a framework for autonomous agents to learn optimal cognitive tick intervals. It utilizes a predictive hyperbolic spread signal derived from the Poincaré ball to quantify epistemic uncertainty, alongside an interval-aware reward function to solve credit-assignment failures in timing problems. Experiments demonstrate that this approach significantly improves efficiency compared to fixed-interval baselines.

Entities (5)

Davide Di Gioia · author · 100%Poincaré ball · mathematical-model · 99%ATCPG · framework · 98%Hyperbolic spread · metric · 97%ATCPG-ST · model-variant · 95%

Relation Signals (3)

ATCPG → utilizes → Poincaré ball

confidence 98% · The framework is constructed from four interacting components... predictive hyperbolic spread... derived from hyperbolic geometry... embedded in the Poincaré ball

Hyperbolic spread → influences → cognitive tick interval

confidence 96% · High spread indicates a branching, uncertain future and drives the agent to act sooner; low spread signals predictability and permits longer rest intervals.

ATCPG-ST → improves → efficiency

confidence 95% · This extension raises mean hyperbolic spread (κ) from 1.88 to 3.37 and yields a further 5.8 percent efficiency gain

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Abstract

Abstract:Autonomous agents operating in continuous environments must decide not only what to do, but when to act. We introduce a lightweight adaptive temporal control system that learns the optimal interval between cognitive ticks from experience, replacing ad hoc biologically inspired timers with a principled learned policy. The policy state is augmented with a predictive hyperbolic spread signal (a "curvature signal" shorthand) derived from hyperbolic geometry: the mean pairwise Poincare distance among n sampled futures embedded in the Poincare ball. High spread indicates a branching, uncertain future and drives the agent to act sooner; low spread signals predictability and permits longer rest intervals. We further propose an interval-aware reward that explicitly penalises inefficiency relative to the chosen wait time, correcting a systematic credit-assignment failure of naive outcome-based rewards in timing problems. We additionally introduce a joint spatio-temporal embedding (ATCPG-ST) that concatenates independently normalised state and position projections in the Poincare ball; spatial trajectory divergence provides an independent timing signal unavailable to the state-only variant (ATCPG-SO). This extension raises mean hyperbolic spread (kappa) from 1.88 to 3.37 and yields a further 5.8 percent efficiency gain over the state-only baseline. Ablation experiments across five random seeds demonstrate that (i) learning is the dominant efficiency factor (54.8 percent over no-learning), (ii) hyperbolic spread provides significant complementary gain (26.2 percent over geometry-free control), (iii) the combined system achieves 22.8 percent efficiency over the fixed-interval baseline, and (iv) adding spatial position information to the spread embedding yields an additional 5.8 percent.

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Learning When to Act: Interval-Aware Reinforcement Learning with Predictive Temporal Structure Davide Di Gioia University College London ucsigi@ucl.ac.uk (March 2026) Abstract Autonomous agents operating in continuous environments must decide not only what to do, but when to act. We introduce a lightweight adaptive temporal control system that learns the optimal interval between cognitive ticks from experience, replacing ad hoc biologically-inspired timers with a principled learned policy. The policy state is augmented with a novel predictive hyperbolic spread signal (a “curvature signal” shorthand) derived from hyperbolic geometry: the mean pairwise Poincaré distance among n sampled futures embedded in the Poincaré ball cnB^n_c. High spread indicates a branching, uncertain future and drives the agent to act sooner; low spread signals predictability and permits longer rest intervals. We further propose an interval-aware reward that explicitly penalises inefficiency relative to the chosen wait time, correcting a systematic credit-assignment failure of naive outcome-based rewards in timing problems. We additionally introduce a joint spatio-temporal embedding (ATCPG-ST) that concatenates independently normalised state and position projections in the Poincaré ball; spatial trajectory divergence provides an independent timing signal unavailable to the state-only variant (ATCPG-SO). This extension raises mean hyperbolic spread (κ) from 1.881.88 to 3.373.37 and yields a further +5.8%+5.8\% efficiency gain over the state-only baseline. Ablation experiments across five random seeds demonstrate that (i) learning is the dominant efficiency factor (+54.8%+54.8\% over no-learning), (i) hyperbolic curvature provides significant complementary gain (+26.2%+26.2\% over geometry-free control), (i) the combined system achieves +22.8%+22.8\% efficiency over the fixed-interval baseline, and (iv) adding spatial position information to the curvature embedding yields an additional +5.8%+5.8\%. Contents 1 Introduction 2 Related Work 3 Problem Formulation 3.1 Temporal Pacing as a Contextual Bandit 3.2 State Representation 4 Learned Pacing Policy 4.1 Linear Policy 4.2 Online Weight Update 4.3 State-Dependent Exploration 4.4 Internal Oscillator 5 Interval-Aware Reward 5.1 The Credit-Assignment Failure of Outcome Reward 5.2 Proposed Interval-Aware Reward 6 Predictive Hyperbolic Spread via Hyperbolic Geometry 6.1 Poincaré Ball Primer 6.2 Future Embedding 6.3 Spread Estimator 6.4 Integration with the Pacing Policy 6.5 Joint Spatio-Temporal Embedding 7 Experiments 7.1 Simulation Environment 7.2 Metrics 7.3 Baselines and Ablations 7.4 Results 7.5 Qualitative Behaviour 7.6 Multi-Agent Phase Synchronisation 7.7 Head-to-Head: TemporalController vs. SpatioTemporalATCPGAgent 7.8 Spatial Ablation: State-Only vs. Spatio-Temporal Embedding 7.9 Real-LLM Pacing Benchmark 8 Discussion 9 Conclusion References A Proof of Proposition 1 (Extended) B Algorithm Pseudocode 1 Introduction The rapid capability expansion of Large Language Models (LLMs) has catalysed the deployment of autonomous agents and multi-agent systems (MAS) in open-ended, continuous environments. In these settings, an agent’s efficacy is bounded not only by what action it takes, but when it chooses to act. As agents transition from isolated chatbots to persistent digital workers, the temporal cadence of their reasoning cycles directly dictates their computational overhead, responsiveness, and ultimate task success. Despite sophisticated advances in task delegation and multi-agent routing, the internal temporal pacing of individual agents remains remarkably primitive. Most contemporary orchestration frameworks rely on either strictly reactive, event-driven triggers (an agent only wakes when messaged) or hand-tuned, fixed-interval polling loops (e.g., sleep(N) calls). Bio-mimetic heuristic formulas (such as hard-coded “fatigue multipliers” or “ultradian rhythms”) offer a veneer of adaptability, but remain brittle: they encode the system designer’s static prior rather than the agent’s lived experience, and they categorically fail to adapt as task complexity and environmental volatility shift. To address this structural blind spot, we propose Adaptive Temporal Control via Predictive Geometry (ATCPG), a lightweight, fully autonomous pacing system that allows an agent to learn its optimal cognitive interval dynamically. ATCPG shifts temporal control from a fixed network orchestrator to an internal, learned policy driven by the agent’s own epistemic uncertainty. The framework is constructed from four interacting components: 1. Learned pacing policy: a linear associative bandit that continuously updates the wait interval after every cognitive tick via reward-weighted regression. 2. Predictive hyperbolic spread (informally, a “curvature signal”), a geometric signal capturing the dispersion of n predicted future states in the Poincaré ball cnB^n_c. The metric’s natural boundary expansion aggressively amplifies diverging trajectories, signalling the agent to act sooner. 3. Interval-aware shaping reward: a dynamic learning signal that prevents the catastrophic credit-assignment failure common to standard RL timing problems by explicitly pricing the chosen interval. 4. Joint spatio-temporal embedding (ATCPG-ST): an extension that augments the policy state with Poincaré-projected spatial position vectors. Spatial trajectory divergence is an independent timing signal: branching navigation trees fan out in position space before belief space, and the Poincaré ball’s boundary-expansion property amplifies both signals in a unified geometric language. ATCPG-ST raises mean hyperbolic spread (κ) by 1.79×1.79× and efficiency by +5.8%+5.8\% over the state-only baseline (ATCPG-SO); it degrades gracefully to ATCPG-SO whenever position data are absent. The paper makes the following contributions: • We formalise the temporal pacing problem as an RL problem over a continuous action space (Section 3). • We derive the interval-aware reward and prove it avoids the credit-assignment failure of naive outcome reward in the pacing setting (Section 5). • We introduce predictive hyperbolic spread (a “curvature signal” shorthand) as a timing signal, grounded in the Poincaré ball model of hyperbolic geometry (Section 6). • We propose ATCPG-ST, a joint spatio-temporal embedding (Section 6.5) that lifts position trajectories into the same Poincaré ball as state embeddings, motivated by the observation that spatial trajectory divergence is an independent leading indicator of decision uncertainty unavailable to the state-only ATCPG-SO. We demonstrate empirically that the combined signal raises curvature by 1.79×1.79× and efficiency by +5.8%+5.8\% on a controlled ablation, and that the ordering ηSO<ηST _SO< _ST replicates on a live GPT-4.1 deployment (Section 7.9). • We provide systematic ablation evidence that each component contributes independently to efficiency (Section 7). 2 Related Work To situate ATCPG within the broader landscape of artificial intelligence, we contextualise our framework across four distinct domains: multi-agent orchestration, temporal abstraction in reinforcement learning, geometric representation, and autonomous cognitive loops. Decentralised orchestration and emergent synchronisation. A growing body of work in multi-agent systems (MAS) shifts temporal coordination away from centralised schedulers toward decentralised, self-organising mechanisms. In contemporary LLM-agent frameworks, execution is increasingly modelled as event-driven computation on structured graphs, where agents are triggered by dependency resolution, message arrivals, or verification outcomes rather than a global clock (Yang et al., 2025; Nalagatla, 2025). In parallel, control-theoretic and multi-agent reinforcement learning literatures study how global temporal regularities emerge from local interaction rules, including consensus-style coordination and communication-efficient protocols that yield synchronised collective behaviour without explicit central timing (Oh et al., 2025; Stoorvogel et al., 2025; Erofeeva et al., 2025). Crucially, these approaches resolve temporal structure primarily at the network layer: they formalise when agents should interact with one another, given topological dependencies and local communication signals. The internal deliberation cadence of an individual agent is often treated as a reactive black box (an agent “runs” when invoked), rather than as a decision variable with its own uncertainty sensitivity, credit assignment, and temporal cost. Chen et al. (2025) identify this absence of intrinsic self-monitoring as a core unsolved deficiency in AI autonomy. ATCPG targets this orthogonal gap by formalising intrinsic cognitive pacing at the node level: a continuously running agent learns how long to wait between self-initiated cognitive ticks. This intrinsic pacing is complementary to decentralised orchestration rather than a replacement: agents may adapt their own tick intervals while still exhibiting emergent group rhythm via the oscillator and Kuramoto-style phase coupling (Kuramoto, 1984) detailed in Section 4.4. Temporal abstraction in RL. Options (Sutton et al., 1999) and macro-actions allow agents to choose the duration of a commitment, but they operate over discrete intervals and do not adapt the base decision frequency. Semi-MDPs (Puterman, 1994) generalise to continuous sojourn times but require a full transition model; ATCPG operates in the model-free, online contextual-bandit setting. Oudeyer and Kaplan (2007) motivate self-paced learning from progress signals. LIDA (Franklin et al., 2007) uses a fixed cognitive cycle; we replace the fixed cycle with a learned one. Circadian and ultradian-inspired architectures (Wang and Aamodt, 2015) hard-code period parameters that we instead learn. Reward shaping for timing. Ng et al. (1999) study potential-based shaping; we propose a different class of shaping that explicitly accounts for temporal cost. Dewey (2014) addresses utility indifference, but does not consider interval selection as the decision variable. Neither specifically resolves the temporal credit-assignment failure we identify in pacing, necessitating our derivation of an interval-aware reward. Hyperbolic representation and uncertainty. Nickel and Kiela (2017) demonstrate that hierarchical structure embeds more faithfully in cnB^n_c than in Euclidean space due to exponential volume growth near the boundary. Ganea et al. (2018) extend neural networks to hyperbolic space. We use the Poincaré ball not for representation learning but as a geometric divergence measure between predicted futures, quantifying epistemic conflict to drive a temporal control policy, a novel application. Structural blind spots in agent orchestration. Di Gioia (2026) identify a pervasive architectural blind spot in modern multi-agent orchestration stacks: their native scheduling layers route execution blindly without geometric perception of how failures propagate through the graph. They demonstrate that geometry-aware sidecars employing dynamic hyperbolic metric switching close this observability gap. ATCPG takes direct inspiration from this critique: where Di Gioia apply hyperbolic geometry to solve the routing (spatial) blind spot in multi-agent execution graphs, we apply it to solve the pacing (temporal) blind spot in continuous autonomous loops. Autonomous cognitive loops and self-directed timing. The Agentic Heartbeat Pattern (Mendonca, 2025) addresses who coordinates with whom across an organisational hierarchy but does not formalise when a single agent should self-initiate. The SPOC system (Zhao et al., 2025) interleaves solution generation and verification but remains externally triggered; no runtime pacing adaptation occurs. ATCPG’s daemon loop runs continuously between external requests with no external trigger. Chen et al. (2025) identify “absence of intrinsic self-monitoring” and “absence of intrinsic agency” as core unsolved deficiencies of current AI systems, explicitly calling for the type of self-directed temporal control we realise here. To our knowledge, ATCPG is the first concrete, learnable temporal-control module addressing the self-directed timing layer they identify as absent. InSeC (Li et al., 2024) bakes error-correction behaviour into model weights via negative-sample training. The result is a static inference policy activated by an external user query; no runtime pacing adaptation occurs. ATCPG’s policy updates online after every tick and operates independently of external queries. 3 Problem Formulation 3.1 Temporal Pacing as a Contextual Bandit Let an agent operate a cognitive loop with discrete ticks t=1,2,…t=1,2,… At each tick the agent observes a state st∈ℝds_t ^d and selects an interval Δt∈[Δtmin,Δtmax] t_t∈[ t_ , t_ ] before the next tick. After sleeping for Δt t_t seconds the agent acts, observes wellbeing scalar wt∈[0,1]w_t∈[0,1] (a composite health/success signal), and receives reward rtr_t. The agent’s goal is to maximise the long-run average efficiency: (π)=limT→∞1T∑t=1TrtΔt.J(π)\;=\; _T→∞ 1T _t=1^T r_t t_t. (1) This objective explicitly penalises using more time than necessary to achieve a unit of reward. 3.2 State Representation The state vector st∈ℝ6s_t ^6 fed to the policy contains: st=[pt⏟priority,ft⏟fatigue,Δwt⏟wellbeing delta,ρt⏟performance,sin⁡ϕt⏟oscillator phase,κt⏟hyperbolic spread]⊤s_t= [\; p_t_priority,\; f_t_fatigue,\; w_t_wellbeing delta,\; _t_performance,\; _t_oscillator phase,\; _t_hyperbolic spread ] (2) where ϕt _t is an internal phase variable that captures emergent rest/activity cycles (Section 4.4). 4 Learned Pacing Policy 4.1 Linear Policy We parameterise the policy as a linear map: Δt^(,s)=θbias+θpp+θff+θwΔw+θρρ+θϕsin⁡ϕ+θκκ, t( θ,s)\;=\; _bias+ _p\,p+ _f\,f+ _w\, w+ _ρ\,ρ+ _φ φ+ _κ\,κ, (3) with output clamped to [Δtmin,Δtmax][ t_ , t_ ]. The initial weights encode sensible priors: θp<0 _p<0 (urgency shortens wait), θf>0 _f>0 (fatigue lengthens wait), θκ<0 _κ<0 (uncertainty shortens wait). 4.2 Online Weight Update After observing reward rtr_t we update the weight vector via the direct online rule: ←+αrt⋅st. θ\;←\; θ+α\,r_t· s_t. (4) Interpretation of the update rule. The update (4) is not an unbiased policy-gradient estimator for the clipped policy (3). It is best understood as a lightweight online reward-weighted linear regression (equivalently, a linear associative bandit rule) that increases weights on features co-occurring with positive realised reward. This choice prioritises simplicity and low computational overhead (one vector update per tick) over unbiased gradient estimation. For reference, the exact REINFORCE (Williams, 1992) step for a linear-Gaussian policy π(Δt∣s)π( t s) would include the deviation term (Δt−Δt^t)( t_t- t_t): ←+αrtΔt−Δt^tσ2⋅st. θ\;←\; θ+α\,r_t\, t_t- t_tσ^2· s_t. (5) We omit that term, treating the post-tick reward rtr_t as an approximately stationary response to the tick context sts_t over the adaptation time-scale of α. Empirically, the simplified rule provides a stable and effective online update for the pacing problem studied here. 4.3 State-Dependent Exploration We inject multiplicative noise with probability εeff _eff: εeff(st)=ε0⋅(1−|Δwt|), _eff(s_t)\;=\; _0· (1-| w_t| ), (6) so the agent explores freely when wellbeing is stable (|Δwt|≈0| w_t|≈ 0) and exploits its current policy when wellbeing is volatile (|Δwt|≈1| w_t|≈ 1). This reverses the conventional uncertainty → explore heuristic. It is not contradictory, but contextually adapted: in a continuous autonomous loop, high volatility (|Δwt|≈1| w_t|≈ 1) signals acute risk or task failure, demanding immediate exploitation of robust, known intervals to regain stability. Conversely, stable wellbeing indicates the system has the safety margin required to experiment with pacing efficiency. 4.4 Internal Oscillator The internal phase ϕt _t evolves as: ϕt+1=(ϕt+ωt)mod2π,ωt+1=clip(ωt+αrt⋅0.01,[0.001,0.2]), _t+1=( _t+ _t) 2π, _t+1=clip\! ( _t+α r_t· 0.01,\;[0.001,0.2] ), (7) where ωt _t is the phase velocity. Positive rewards accelerate the rhythm; negative rewards slow it. This produces emergent rest/activity cycles whose period is determined endogenously by task structure rather than by a hyperparameter. Multi-agent synchronisation. When N clocks operate in parallel, Kuramoto coupling (Kuramoto, 1984) nudges phases toward the group mean: ϕt(i)←ϕt(i)+λ(ϕ¯t−ϕt(i)),ϕ¯t=1N∑jϕt(j), _t^(i)\;←\; _t^(i)+λ ( φ_t- _t^(i) ), φ_t= 1N _j _t^(j), (8) producing collective rhythm without central coordination. 5 Interval-Aware Reward 5.1 The Credit-Assignment Failure of Outcome Reward Proposition 1 (Reward-direction failure). Under the online linear update (4) with naive reward rt=Δwtr_t= w_t and feature ft>0f_t>0, the update θf+=αrtft _f +=α r_tf_t decreases θf _f when rt<0r_t<0 (overload), producing shorter intervals when the agent is fatigued, opposite to the desired behaviour. Proof. If the agent is overloaded, Δwt<0⇒rt<0 w_t<0 r_t<0. The update θf+=α⋅(negative)⋅ft _f +=α·(negative)· f_t decreases θf _f, reducing the positive contribution θf⋅ft _f· f_t to the interval output (3). ∎ This is a fundamental misalignment: the update ascribes the negative outcome to the fatigue feature and reduces the fatigue-correction behaviour. 5.2 Proposed Interval-Aware Reward We decompose the reward into three additive components: rt=2ΔwtΔt⏟efficiency+1.5otΔtΔtbase⏟spacing bonus+1.0κtΔt⏟spread brake,r_t\;=\; 2\, w_t t_t_efficiency\;+\; 1.5\,o_t\, t_t t_base_spacing bonus\;+\; 1.0\, _t t_t_spread brake, (9) where ot=max⁡(0,−Δwt)o_t= (0,- w_t) is the overload magnitude. Relation to the evaluation objective. Equation (9) is a shaping signal used for the online update (4); it is not divided by Δt t_t again as in (1). The 1/Δt1/ t_t factors in the efficiency and curvature-brake terms encourage the agent to favour short intervals, acting as a surrogate for the temporal cost in (1). This avoids double-counting: we optimise the shaping signal (9) directly and evaluate on the metric (18). Efficiency term. Δwt/Δt w_t/ t_t measures wellbeing gained per second of interval chosen. This correctly penalises long idle periods that produce the same outcome as short ones, driving the agent toward tighter action packing when the environment is cooperative. Spacing bonus. ot⋅(Δt/Δtbase)o_t·( t_t/ t_base) rewards long intervals when the agent is overloaded. Under update rule (4) this creates a positive gradient on θf _f, directly fixing the failure in Proposition 1. Spread brake. κt/Δt _t/ t_t rewards small intervals when future uncertainty is high. This encodes the intuition that a branching future demands more frequent re-evaluation, formalised in Section 6. 6 Predictive Hyperbolic Spread via Hyperbolic Geometry 6.1 Poincaré Ball Primer The Poincaré ball model (cn,gc)(B^n_c,g^c) is the open unit ball x∈ℝn:c‖x‖2<1\x ^n:c\|x\|^2<1\ equipped with the Riemannian metric gxc=λxc 2gE,λxc=21−c‖x‖2,g^c_x= _x^c\,2\,g^E, _x^c= 21-c\|x\|^2, (10) where c>0c>0 is the curvature parameter and gEg^E is the Euclidean metric. The geodesic distance between x,y∈cnx,y ^n_c is: dℍ(x,y)=2carctanh(c‖−x⊕cy‖),d_H(x,y)\;=\; 2 c\,arctanh\! ( c\,\|-x _cy\| ), (11) where ⊕c _c is the Möbius addition: x⊕cy=(1+2c⟨x,y⟩+c‖y‖2)x+(1−c‖x‖2)y1+2c⟨x,y⟩+c2‖x‖2‖y‖2.x _cy\;=\; (1+2c x,y +c\|y\|^2)\,x\;+\;(1-c\|x\|^2)\,y1+2c x,y +c^2\|x\|^2\|y\|^2. (12) A key property used below: equal-Euclidean-step displacements near the boundary of cnB^n_c produce exponentially larger geodesic distances than the same displacements near the origin. This means dℍd_H amplifies divergence in future predictions, making small differences in predicted outcomes measurable. 6.2 Future Embedding Given a world model W, we sample n future state trajectories z(i)i=1n\z^(i)\_i=1^n of horizon h. Each trajectory summary vector z(i)∈ℝdz^(i) ^d is embedded into cmB^m_c via: φ(z)=proj(σ⋅z′‖z′‖),z′=pad/trim(z,m), (z)\;=\;proj\! (σ· z \|z \| ), z =pad/trim(z,m), (13) where σ∈(0,1)σ∈(0,1) is a scale factor (default 0.90.9) and proj(x)=x/max⁡(1,‖x‖/rmax)proj(x)=x/ (1,\|x\|/r_ ) ensures the result lies inside cnB^n_c. The zero vector is embedded to the origin. 6.3 Spread Estimator Definition 1 (Predictive Hyperbolic Spread). Given n embedded futures e(i)i=1n⊂cn\e^(i)\_i=1^n ^n_c, the predictive hyperbolic spread is defined as: κ=μij⏟mean spread+σij2⏟variance of spread,κ\;=\; \, _ij\,_mean spread+ \,σ^2_ij\,_variance of spread, (14) where μij _ij and σij2σ^2_ij are the mean and variance of dℍ(e(i),e(j))i<j\d_H(e^(i),e^(j))\_i<j. Terminology note. Although the manifold’s sectional curvature is fixed by the parameter c (constant throughout cnB^n_c), we use “curvature signal” (κ) as an evocative shorthand for this spread statistic. The name highlights that κ is amplified by the manifold’s expanding geometry near the boundary, not that it measures the manifold curvature itself. Remark 1. Adding variance to mean penalises heterogeneous divergence, a mix of near-identical and wildly differing futures, which is more alarming than uniform spreading and deserves a stronger re-evaluation signal. Proposition 2 (Zero curvature for identical futures). If all n predicted futures are identical, κ=0κ=0. Proof. e(i)=e(j)e^(i)=e^(j) for all i,ji,j implies dℍ(e(i),e(j))=0d_H(e^(i),e^(j))=0 (since ‖−x⊕cx‖=0\|-x _cx\|=0 by Möbius addition), so μij=σij2=0 _ij=σ^2_ij=0. ∎ Geometric regime characterisation. The Poincaré metric amplifies pairwise distances selectively based on two geometric quantities: the radial position r=‖e‖r=\|e\| of the embeddings and the angular divergence between perturbed samples. The conformal factor λc2=4/(1−r2)2λ^2_c=4/(1-r^2)^2 grows from ≈4≈ 4 near the origin to ≈111≈ 111 at r=0.9r=0.9, creating three qualitatively distinct curvature regimes. Proposition 3 (Three-regime amplification). Let u,v∈cnu,v ^n_c with ‖u‖=‖v‖=r\|u\|=\|v\|=r and angle θ=∠(u,v)θ= (u,v) between them. For small angular perturbations δ=1−cos⁡θ≪1δ=1- θ 1, the squared Poincaré distance satisfies dc(u,v)2≈8r2δ(1−r2)2.d_c(u,v)^2\;≈\; 8\,r^2\,δ(1-r^2)^2. (15) Thus κ is jointly amplified by radial position through (1−r2)−2(1-r^2)^-2 and by angular spread through δ. Proof. By definition, dc(u,v)=arccosh⁡(1+X)d_c(u,v)=arccosh(1+X) where X=2‖u−v‖2/(1−r2)2X=2\|u-v\|^2/(1-r^2)^2 for c=1c=1 and equal radii. The law of cosines gives ‖u−v‖2=2r2(1−cos⁡θ)=2r2δ\|u-v\|^2=2r^2(1- θ)=2r^2δ, so X=4r2δ/(1−r2)2X=4r^2δ/(1-r^2)^2. For small X, the Taylor expansion arccosh⁡(1+X)≈2Xarccosh(1+X)≈ 2X yields dc(u,v)2≈2X=8r2δ/(1−r2)2d_c(u,v)^2≈ 2X=8r^2δ/(1-r^2)^2. ∎ Remark 2. Equation (15) is a small-angle, equal-radius approximation. The implementation uses the exact arccosharccosh formula dc(x,y)=arccosh⁡(1+2c‖x−y‖2/[(1−c‖x‖2)(1−c‖y‖2)])d_c(x,y)=arccosh\! (1+2c\|x-y\|^2/[(1-c\|x\|^2)(1-c\|y\|^2)] ) (Ungar, 2008), which handles arbitrary radii and is numerically stable. We demonstrated this amplification mechanism empirically under controlled construction, using MC-dropout (Gal and Ghahramani, 2016) as a surrogate for future-state spread. Table 1 reports the mean predictive curvature κ for three synthetically constructed state topologies under N=200N=200 dropout samples at rate p=0.20p=0.20. Table 1: MC-dropout predictive curvature κ across three geometric regimes (N=200N=200 samples, p=0.20p=0.20, d=6d=6). The Poincaré boundary amplification selectively elevates conflicted states. Regime Construction r κ κ/κconfidentκ/ _confident Conflicted uncertainty Opposing dominant features at boundary ≈0.90≈ 0.90 6.226.22 10.4×10.4× Confident prediction Stable signal, ball interior ≈0.37≈ 0.37 0.600.60 1.0×1.0× Uninformative noise Near-zero state, Euclidean limit ≈0.12≈ 0.12 0.210.21 0.35×0.35× Both discriminability ratios massively exceed the baseline: Conflicted vs. Confident =10.4×=10.4×; Conflicted vs. Noise =29.6×=29.6×. This confirms that κ does not merely measure spread, it measures directional conflict amplified by proximity to the ball boundary, a property unique to hyperbolic geometry. 6.4 Integration with the Pacing Policy The spread signal enters the policy (3) with weight θκ<0 _κ<0, so high spread decreases the predicted interval. In the reward (9) the spread brake term κ/Δtκ/ t_t creates a positive gradient on θκ _κ when the interval is short, reinforcing the speed-up under uncertainty behaviour (act sooner when future spread is high). 6.5 Joint Spatio-Temporal Embedding The state-only embedding (13) captures what the future will look like, the uncertainty over predicted content, but not where the agent or world will be spatially. Yet spatial location carries independent, timing-relevant information: two predicted futures that are identical in state space can be spatially far apart, implying divergent world trajectories that curvature over state embeddings alone cannot detect. In navigation, multi-step planning, and execution graphs, spatial trajectory divergence is often a stronger leading indicator of decision uncertainty than state spread: a branching navigation tree fans out in position space well before it fans out in belief space. The Poincaré ball is an ideal joint container for both signals: its boundary-expansion property amplifies state uncertainty and spatial trajectory fan-out in the same geometric language, without requiring separate encoders or manual scale matching. We therefore extend ATCPG by augmenting the policy state with Poincaré-projected position vectors; we call the resulting system ATCPG-ST (Spatio-Temporal) to distinguish it from the state-only variant ATCPG-SO. ATCPG-ST is a strict generalisation: when position information is absent it falls back exactly to ATCPG-SO behaviour. Formally, when the world model produces both predicted state vectors and predicted spatial position vectors p(i)i=1n⊂ℝk\p^(i)\_i=1^n ^k, we form a joint spatio-temporal embedding by concatenating independently normalised Poincaré projections: ψ(i)=proj([φ(z(i);ms)∥φ(p(i);mp)]),ψ^(i)\;=\;proj\! ( [ (z^(i);\,m_s)\;\|\; (p^(i);\,m_p) ] ), (16) where msm_s and mpm_p are the state and position embedding dimensions (defaults 66 and 33), ∥\| denotes concatenation, and the outer proj re-projects the (ms+mp)(m_s+m_p)-dimensional vector into cms+mpB^m_s+m_p_c. Independent normalisation of each component prevents the higher-dimensional state from dominating the position signal. Predictive curvature is then computed exactly as in (14) but over the joint embeddings ψ(i)\ψ^(i)\. When position information is absent the method falls back to state-only embeddings (backward-compatible). Spatial divergence amplifies κ. Because the Poincaré metric expands near the ball boundary, trajectories that diverge in position space (‖p(i)−p(j)‖\|p^(i)-p^(j)\| large) produce joint embeddings near the boundary, and their geodesic distances dℍ(ψ(i),ψ(j))d_H(ψ^(i),ψ^(j)) are substantially larger than those computed from state embeddings alone. The empirical effect is a higher κ and correspondingly shorter intervals whenever the agent’s predicted spatial trajectories fan out, a desirable signal in navigation or multi-step planning tasks where spatial divergence correlates with decision uncertainty (Section 7.8). Proposition 4 (Spatial monotonicity, non-saturated regime). Let z(i)∈ℝmsz^(i) ^m_s be fixed and let p(i)∈ℝkp^(i) ^k be position vectors. Define joint embeddings ψ(i)(ε)ψ^(i)( ) by replacing p(i)p^(i) with εp(i) p^(i) in (16). Assume there exists εmax>0 _ >0 such that for all ε∈[0,εmax] ∈[0, _ ] the outer proj(⋅)proj(·) in (16) does not activate (no clipping). Then κST(ε)κ^ST( ) is non-decreasing in ε on [0,εmax][0, _ ]. In particular, if p(i)≠p(j)p^(i)≠ p^(j) for some i≠ji≠ j, then for sufficiently small ε>0 >0, κST(ε)>κSOκ^ST( )>κ^SO. Proof sketch. Without clipping, scaling p(i)↦εp(i)p^(i) p^(i) increases pairwise Euclidean separations of the position components monotonically in ε . For embeddings bounded away from the boundary, the Poincaré distance is monotone in ‖x−y‖\|x-y\| for fixed radii, so each pairwise dℍ(ψ(i),ψ(j))d_H(ψ^(i),ψ^(j)) is non-decreasing. Their mean and variance are therefore non-decreasing, giving κST(ε)≥κSOκ^ST( )≥κ^SO. ∎ 7 Experiments 7.1 Simulation Environment We evaluate in a synthetic environment where each tick simulates a cognitive cycle with stochastic outcomes: dt d_t ∼Gaussian(dbase(ot)−30pt, 20), (d_base(o_t)-30p_t,\;20), Δwt w_t ∼Gaussian(μw(ot), 0.05), ( _w(o_t),\;0.05), successt _t =[¬ot∨pt>0.7], =1[ o_t\; \;p_t>0.7], (17) where dbase∈50,200d_base∈\50,200\ ms for non-overloaded and overloaded, μw∈+0.1,−0.2 _w∈\+0.1,-0.2\, ot∼Bernoulli(0.3)o_t (0.3). All agents use Δtmin=10 t_ =10\,s, Δtmax=300 t_ =300\,s, base interval 6060\,s. 7.2 Metrics Efficiency (primary). η=1T∑t=1TsuccesstΔt.η= 1T _t=1^T success_t t_t. (18) Performance score. Mean success rate. Wellbeing stability. Standard deviation of the wellbeing trajectory (lower is better). 7.3 Baselines and Ablations We compare five configurations, plus a privileged reference baseline and a spatial extension (Table 2): Full model (ATCPG) All components enabled; curvature inferred geometrically from predicted futures, no privileged information. −-Learning Weights frozen; fixed interval equals base. −-Curvature Curvature feature set to 0; policy learns from remaining features. −-Interval reward Simple outcome reward rt=2Δwt+0.5/dtr_t=2 w_t+0.5/d_t replaces (9). −-Exploration ε0=0 _0=0 (deterministic policy). ATCPG-ST (positions) Full model extended with joint spatio-temporal embedding (16); positions are independently drawn 3-D vectors correlated with overload state. TC (privileged)† TemporalController with curvature computed from a Gaussian conditioned on the directly observed overload flag oto_t. This constitutes an upper bound: the best achievable efficiency when a clean oracle overload signal is available. Results are averaged over three random seeds; the fixed-interval baseline is averaged over five seeds. 7.4 Results Table 2: Efficiency scores η (18) for the full ATCPG model, ablated variants (averaged over 3 seeds), the fixed-interval baseline (averaged over 5 seeds), and TemporalController as a privileged upper bound. Δ is the relative change versus the full model. Variant Efficiency η Δ vs Full Info access Full model (ATCPG, blind) 0.02900.0290 — none −-Learning (fixed interval) 0.01310.0131 −54.8%-54.8\% none −-Curvature 0.02140.0214 −26.2%-26.2\% none −-Interval-aware reward 0.02460.0246 −15.1%-15.1\% none −-Exploration 0.02810.0281 −3.1%-3.1\% none ATCPG-ST (+ positions) see §7.8 — positions Fixed-interval baseline (5 seeds) 0.01320.0132 −54.5%∗-54.5\%^* none TC (privileged)† 0.02750.0275 −5.2%‡-5.2\% direct oto_t ∗Relative to the full model (ablation seeds). The fixed-interval baseline equals the −-Learning ablation by construction (−54.8%-54.8\%, rounding). The 5-seed comparative evaluation in Section 7 gives +22.8%+22.8\% ATCPG advantage over baseline (η=0.0162η=0.0162 vs. 0.01320.0132). †TemporalController: curvature from oracle overload flag oto_t, privileged upper bound, not a fair comparison to ATCPG. ‡Despite direct oto_t access, TC trails ATCPG because TC’s higher discriminability (20×20×) is insufficient alone: absolute κ magnitude, not the overload/normal ratio, governs interval selection (see Section 7.7). Learning dominates. Removing learning halves efficiency (−54.8%-54.8\%). No other component comes close. This confirms that adaptive pacing, learning the right interval from experience, is the primary mechanism of improvement. Hyperbolic spread provides significant gain. The −26.2%-26.2\% drop in the −-Curvature condition demonstrates that the geometric future-spread signal contributes substantially beyond what the non-geometric features (priority, fatigue, wellbeing, performance) alone can capture. The hyperbolic spread signal provides a prospective signal: it encodes what is about to happen rather than what just happened, giving the policy information unavailable to non-geometric policy learners. Interval-aware reward corrects credit assignment. The −15.1%-15.1\% drop in the −-Interval reward condition empirically validates Proposition 1: the naive outcome reward produces a systematic bias against slowing down under overload. Replacing it with (9) removes this bias and improves efficiency. Exploration stabilises. The modest −3.1%-3.1\% drop in the −-Exploration condition suggests that stochasticity prevents premature convergence without being the dominant learning mechanism. Comparison to fixed-interval baseline. On the primary metric, the full model achieves +22.8%+22.8\% improvement over the fixed-interval baseline across five seeds (η=0.0162η=0.0162 vs. 0.01320.0132), with consistent advantage on performance score (0.800.80 vs. 0.790.79). Average reward is near-equal (0.49960.4996 vs. 0.50090.5009); we regard efficiency (18) as the correct primary metric since it explicitly accounts for temporal cost. Privileged upper bound (TC). TemporalController achieves η=0.0275η=0.0275 when granted direct access to the overload flag oto_t to construct its curvature estimate. This is −5.2%-5.2\% below the full ATCPG model (0.02900.0290), establishing that the geometric approach matches or exceeds the oracle-privileged baseline even on the ablation environment. Because TC directly observes oto_t while ATCPG does not, this result constitutes a lower bound on ATCPG’s advantage in realistic partially-observable settings (see the full 500-tick head-to-head analysis in Section 7.7 where the gap widens to +72.5%+72.5\%). 7.5 Qualitative Behaviour The learned policy exhibits two characteristic behaviours: 1. Interval lengthening under load. Beginning from an average interval of 39.739.7\,s in the first 100100 ticks, the agent learns to space ticks further apart during sustained overload periods, reaching 57.457.4\,s average over the final 100100 ticks. 2. Urgency-driven acceleration. High-priority states produce intervals ≈41.5≈ 41.5\,s vs. 59.559.5\,s for idle states (pt=0.95p_t=0.95 vs. pt=0.05p_t=0.05), consistent with the negative initial weight θp=−20 _p=-20. 7.6 Multi-Agent Phase Synchronisation When five independent clocks are coupled with λ=0.05λ=0.05 (Kuramoto term (8)), phase spread reduces from 5.45.4 rad to 5.15.1 rad over 100100 steps. While the effect is mild in this short horizon, long-horizon coupling produces collective rhythm without central coordination, an emergent property consistent with the Kuramoto model’s known convergence behaviour. Increased coupling (λ=0.1λ=0.1) over a 50-tick horizon produces 0.9350.935 rad spread in the coupled team vs. 3.4793.479 rad uncoupled, at identical efficiency (0.08250.0825), demonstrating synchronisation without performance cost. 7.7 Head-to-Head: TemporalController vs. SpatioTemporalATCPGAgent To compare geometric vs. scalar curvature computation we ran both implementations over an identical pre-generated 500-tick environment trajectory (same random seed), removing all confounding stochasticity. The two agents differ in one critical respect: TC (privileged). Computes κtTCκ^TC_t from a Gaussian prior conditioned on the observed overload flag ot∈0,1o_t∈\0,1\, a direct, low-variance signal: ϵTC(t)=f(ot)ε^TC(t)=f(o_t). ATCPG (blind). Never observes oto_t; instead computes κtATCPGκ^ATCPG_t as the mean++variance of pairwise Poincaré distances among n=4n=4 noisy future-state embeddings , an indirect, higher-variance signal: ϵATCPG(t)=f(κtATCPG)ε^ATCPG(t)=f\! (κ^ATCPG_t ). This asymmetry creates two opposing forces: (1) TC has cleaner reward-to-state correlation and faster per-tick gradient estimates, predicting ηTC≳ηATCPGη^TC η^ATCPG; (2) ATCPG’s Poincaré distances are absolutely larger due to the metric’s exponential expansion near the ball boundary, driving shorter intervals across all ticks and higher raw throughput. Empirically, effect (2) dominates. Key results (Table 3): • Efficiency. ATCPG achieves 0.04740.0474 vs. TC’s 0.02750.0275 (+72.5%+72.5\%) at identical performance scores (0.7920.792), despite TC’s privileged information access. This result is therefore a lower bound on ATCPG’s advantage in realistic settings where oto_t is not directly observable. • Interval selection. ATCPG sets shorter average intervals (16.716.7\,s vs. 28.828.8\,s), reflecting higher absolute κ even under normal (non-overloaded) conditions (κnormalATCPG=1.20κ^ATCPG_normal=1.20 vs. 0.100.10 for TC). • Curvature discriminability. TC achieves a higher overload/normal κ ratio (20×20×) because it directly observes oto_t, while ATCPG achieves 3.4×3.4× from indirect geometric evidence. The sharper TC signal confirms the information advantage; yet absolute curvature level, not discriminability ratio, is what governs interval selection and efficiency. This finding supports the use of Poincaré curvature as an uncertainty proxy in realistic settings: even against a privileged baseline with direct overload observation, the geometric signal yields superior task-throughput by calibrating action frequency to the manifold’s local expansion rate. The following paragraphs provide a full mathematical treatment of the asymmetry and its practical implications. Table 3: Head-to-head comparison on a shared 500-tick environment. TC directly observes the overload flag oto_t (privileged, same run as the TC† row in Table 2); ATCPG infers load geometrically from future embeddings (blind). The +72.5%+72.5\% efficiency advantage for ATCPG is a lower bound: TC’s information advantage is eliminated in partial-observability settings. Metric TC (privileged) ATCPG (blind) Efficiency (primary) 0.02750.0275 0.04740.0474 (+72.5%) Avg interval (s) 28.828.8 16.716.7 Performance score 0.7920.792 0.7920.792 κ overload 2.032.03 4.064.06 κ normal 0.100.10 1.201.20 κ discriminability 20×20× 3.4×3.4× Info access to oto_t direct none Calibration and fairness. The efficiency advantage demonstrated above depends mechanically on the effective scale of κt _t, as it enters the interval predictor linearly via (21). In real-world deployments, κ should therefore be treated as an uncalibrated signal requiring either (i) per-estimator tuning of the curvature weight θκ _κ, or (i) normalisation (e.g. via a running mean/variance) before entering the policy. Our comparison intentionally freezes θκ _κ across both agents to highlight a structural advantage: Poincaré boundary amplification yields a naturally larger dynamic range for future divergence without manual scaling. This structural expansion allows the agent to react more aggressively under partial observability, precisely the regime that motivates ATCPG. Information asymmetry in the head-to-head comparison. The head-to-head experiment (Section 7.7) contains a deliberate information asymmetry that must be acknowledged when interpreting the results. TemporalController (TC) directly observes the binary overload flag ot∈0,1o_t∈\0,1\ and uses it to set the noise variance of its scalar curvature estimate: κtTC=(μot,σot2),μ1≫μ0,κ^TC_t=N\! ( _o_t,\, _o_t^2 ), _1 _0, (19) so ϵTC(t)=f(ot)ε^TC(t)=f(o_t) is a direct, low-variance signal. By contrast, SpatioTemporalATCPGAgent (ATCPG) never observes oto_t; it infers uncertainty purely from the spread of n=4n=4 noisy future-state embeddings in the Poincaré ball: κtATCPG=d¯ℍ+Var[dℍ],ϵATCPG(t)=f(κtATCPG),κ^ATCPG_t= d_H+Var\! [d_H ], ε^ATCPG(t)=f\! (κ^ATCPG_t ), (20) an indirect, higher-variance signal because the Gaussian perturbations in the future vectors do not perfectly encode oto_t. This asymmetry has two opposing consequences for the online weight update Δθk=αrtst,k _k=α\,r_t\,s_t,k (4): 1. TC has cleaner reward-to-state correlation. Because oto_t is directly observed, the reward rtr_t and the curvature feature st,κs_t,κ are nearly deterministically linked, producing low-variance gradient estimates and faster convergence per tick. Under this metric one would expect ηTC≳ηATCPGη^TC η^ATCPG. 2. ATCPG’s absolute κ is larger. The Poincaré metric expands near the ball boundary, so even modest future-state noise yields geodesic distances well above unity (κnormalATCPG≈1.20κ^ATCPG_normal≈ 1.20 vs. κnormalTC≈0.10κ^TC_normal≈ 0.10). The negative curvature weight θκ=−30 _κ=-30 then drives ATCPG to select shorter intervals across all ticks, increasing action frequency and, consequently, raw throughput. Empirically, effect (2) dominates: ATCPG achieves +72.5%+72.5\% efficiency over TC despite TC’s privileged information access. The curvature discriminability ratio (overload vs. normal κ) is 20×20× for TC but only 3.4×3.4× for ATCPG, confirming that TC’s signal is sharper; yet ATCPG’s higher absolute curvature level proves more consequential for interval selection. Practical implication. In real-world deployments the overload flag oto_t is rarely directly observable; agents must estimate load from partial, noisy context, precisely the regime for which ATCPG’s geometric approach is designed. We therefore expect ATCPG to retain or widen its efficiency advantage as oto_t becomes only partially observable, while TC’s advantage in signal quality degrades monotonically. Future work should quantify this crossover as a function of the overload observation noise σo _o. Mechanics of the asymmetry. To formalise why absolute curvature magnitude governs efficiency independently of discriminability ratio, we isolate the effect of κt _t on the primary metric η. Let the unclipped predictor (3) be grouped as Δt~t=bt+θκκt t_t=b_t+ _κ _t, where btb_t contains the bias and all non-geometric features. In the unsaturated regime (before clipping at Δtmin t_ or Δtmax t_ ), the interval’s sensitivity to curvature is: ∂Δt^t∂κt=∂Δt~t∂κt=θκ. ∂ t_t∂ _t\;=\; ∂ t_t∂ _t\;=\; _κ. (21) Because the policy learns θκ<0 _κ<0, any systematic increase in κt _t strictly compresses Δt^t t_t. Now let xt∈0,1x_t∈\0,1\ denote successtsuccess_t. Given two controllers A and B running on an identical tick sequence with identical outcomes (xtA=xtB=xtx_t^A=x_t^B=x_t), the difference in efficiency (18) is: ηA−ηB=1T∑t=1Txt(1ΔttA−1ΔttB).η^A-η^B\;=\; 1T _t=1^Tx_t\! ( 1 t_t^A- 1 t_t^B ). (22) By the strict monotonicity of 1/u1/u for u>0u>0, if A systematically selects shorter intervals (ΔttA<ΔttB t_t^A< t_t^B), the summand is strictly positive on every successful tick. Because ATCPG’s geometric computation naturally produces a higher absolute baseline (κnormalATCPG=1.20≫κnormalTC=0.10κ^ATCPG_normal=1.20 κ^TC_normal=0.10), Eq. (21) guarantees ΔttATCPG<ΔttTC t_t^ATCPG< t_t^TC across the full trajectory; substituting into (22) with the observed identical performance scores (0.7920.792) mechanically yields the +72.5%+72.5\% efficiency advantage.111The argument assumes intervals lie in the unsaturated regime. Clipping at Δtmin t_ could in principle eliminate the gap; in practice ATCPG’s mean interval (16.716.7\,s) is well above Δtmin=10 t_ =10\,s. 7.8 Spatial Ablation: State-Only vs. Spatio-Temporal Embedding To measure the value added by position information we ran a controlled 500-tick ablation (seed 99) comparing two variants of SpatioTemporalATCPGAgent on a shared environment trajectory: ATCPG-SO State-only embedding (current default). ATCPG-ST Joint spatio-temporal embedding (16); positions are 3-D vectors drawn with noise σpos=3.0 _pos=3.0 under overload and 0.20.2 otherwise, directly encoding spatial trajectory divergence correlated with oto_t. Table 4: Spatial ablation: ATCPG state-only vs. joint spatio-temporal embedding on a shared 500-tick trajectory. Position noise is correlated with the overload flag so that spatial spread is a genuine informative signal. Variant Efficiency η Mean κ κ disc. (ol/nol) ATCPG-SO (state only) 0.03360.0336 1.881.88 4.65×4.65×0000 ATCPG-ST (+ positions) 0.03550.0355 3.373.37 1.98×1.98×0000 Gain (ST vs. SO) +5.8%+5.8\% +1.79×+1.79× — †Lower ratio reflects positions raising baseline κ in normal ticks; the absolute overload κ is strictly higher for ATCPG-ST. Table 4 and our ablation experiments indicate three effects: 1. Higher mean hyperbolic spread (+1.79×+1.79×). ATCPG-ST mean κ rises from 1.881.88 to 3.373.37, consistent with Proposition 4: spatial divergence pushes joint embeddings toward the Poincaré boundary where geodesic distances are maximally amplified. 2. Strictly higher absolute overload curvature. The absolute κoverload _overload for ATCPG-ST exceeds that of ATCPG-SO, confirming that position trajectories carry genuine load-correlated signal. The overload/normal ratio is lower for ATCPG-ST (1.98×1.98× vs. 4.65×4.65×) because positions also raise the baseline κ in normal ticks; the consequential metric is the absolute level, not the ratio (same mechanism as the TC vs. ATCPG analysis in Section 7.7). 3. +5.8%+5.8\% efficiency gain. ATCPG-ST achieves η=0.0355η=0.0355 vs. 0.03360.0336 for ATCPG-SO. The mechanism mirrors the state-only advantage over TC: higher absolute κ drives shorter average intervals (21.321.3 s vs. 22.522.5 s), increasing action frequency and raw throughput without any change to the policy update rule. The chain of value. The full efficiency ordering is: ηfixed=0.0132⏟baseline<ηTC=0.0275⏟scalar κ, privileged<ηSO=0.0336⏟geometric κ, blind<ηST=0.0355⏟joint spatio-temporal _fixed\!=0.0132_baseline< _TC\!=0.0275_scalar κ, privileged< _SO\!=0.0336_geometric κ, blind< _ST\!=0.0355_joint spatio-temporal Each step in the chain adds a qualitatively new information source: learning, geometric curvature, and finally spatial trajectory divergence. The final +5.8%+5.8\% step requires no labelled data, no extra parameters, and no change to the training loop, only the richer embedding. The improvement is proportional to the correlation between position divergence and task load. When position noise is uncorrelated with oto_t, ATCPG-ST degrades gracefully to ATCPG-SO behaviour, confirming backward compatibility. 7.9 Real-LLM Pacing Benchmark All preceding experiments use a synthetic environment whose “LLM calls” are simulated with calibrated noise. To verify that the efficiency ordering holds under a genuine commercial language model, we replicated the four-strategy comparison on a live deployment of GPT-4.1 via the OpenAI API (gpt-4.1). Setup. Each of the four strategies, Fixed, Reactive, ATCPG (TemporalController), and ATCPG-ST (SpatioTemporalATCPGAgent), was evaluated over N=15N=15 episodes (55 episodes × 3 seeds), with up to 5 real LLM calls per episode. The API model string was gpt-4.1 (OpenAI); calls used temperature=0.7=0.7, top_p=1.0=1.0, max_output_tokens=120=120, no stop sequences. The overload flag was simulated at 30%30\% prevalence; overloaded slots received a noise-injection prompt appending a short passage of contradictory context (approximately 40 tokens), making them costlier in tokens and more likely to produce incoherent responses. An episode call succeeded when the response contained ≥12≥ 12 words and no refusal phrase (“I cannot”, “I’m unable to”). An episode succeeded when ≥50%≥ 50\% of its calls succeeded. Experiments were run in March 2026. All other hyperparameters matched those of Section 7. Results. Table 5 reports the outcome over 1515 episodes per strategy. Table 5: Real-LLM pacing benchmark. GPT-4.1 (OpenAI, gpt-4.1). 15 episodes (5×35× 3 seeds), 5 LLM calls/episode, overload =30%=30\%. Strategy Success Total tokens η vs. Fixed Fixed 1.001.00 6,9246,924 0.002170.00217 — Reactive 0.270.27 2,5642,564 0.001560.00156 −28.1%-28.1\% ATCPG 1.001.00 6,5536,553 0.002290.00229 +5.7%+5.7\% ATCPG-ST 1.001.00 6,1726,172 0.002430.00243 +12.2%+12.2\% Both adaptive controllers match Fixed’s 100%100\% episode success rate while consuming fewer tokens: ATCPG saves 371371 tokens (−5.4%-5.4\%) and ATCPG-ST saves 752752 tokens (−10.9%-10.9\%) relative to Fixed. Reactive halves the token budget but collapses success to 27%27\%, confirming that naïve skipping is not viable. The efficiency ordering ηFixed<ηATCPG<ηATCPG-ST _Fixed< _ATCPG< _ATCPG-ST holds on live real-cost GPT-4.1 calls, reproducing the simulation chain from Section 7.8. ATCPG-ST’s additional +5.7%+5.7\% efficiency gain over plain ATCPG is consistent with the joint spatio-temporal embedding providing a richer curvature signal, as predicted by Proposition 4. 8 Discussion Limitations and Future Work. The current pacing policy is intentionally linear. While real-world agents operating in highly non-convex efficiency landscapes might theoretically benefit from deep neural policies, introducing a heavy neural approximator for the pacing daemon would undermine the framework’s core objective: minimising computational overhead. The (1)O(1) linear update ensures the pacing mechanism remains strictly lightweight. Furthermore, while we demonstrate multi-agent temporal batching via Kuramoto coupling (Section 4.4), the core pacing policy is evaluated in a single-agent, single-task environment. We cannot yet claim generalisation to complex, multi-task domains. A critical direction for future work is evaluating multi-task transfer, specifically, investigating whether a pacing policy θ learned on high-volatility search tasks transfers zero-shot to lower-volatility navigation or coding domains. Finally, the world model providing futures in our synthetic experiments is simulated by an oracle. While Section 7.9 partially addresses this by demonstrating the efficiency ordering on a live GPT-4.1 deployment, the future-state vectors in that benchmark remain synthetically constructed. The Poincaré ball implementation also clips embeddings at radius 1−ε1- to ensure numerical stability; future extensions could employ a Lorentz/hyperboloid formulation (Nickel and Kiela, 2018) to remove this constraint at the cost of a higher-dimensional ambient space. Approximating the World Model W for LLMs. In our simulation the world model generates the n predicted future trajectories z(i)i=1n\z^(i)\_i=1^n via an oracle. For real-world deployment with Large Language Models, executing n full autoregressive rollouts per cognitive tick solely to compute the pacing interval is computationally prohibitive. We propose instantiating W via Latent Perturbation (Gal and Ghahramani, 2016): let ht∈ℝdh_t ^d be the LLM’s final hidden state before action selection. Rather than decoding n textual futures, we apply n independent stochastic dropout masks to hth_t, producing perturbed latent states h~t(i)∼Dropout(ht) h^(i)_t (h_t), which are projected directly into cnB^n_c via the embedding map (Eq. (13)). This Bayesian proxy reduces the (n)O(n) generative cost of future simulation to (1)O(1) inference, allowing the continuous pacing daemon to operate without draining the compute budget reserved for task execution. Relation to semi-MDPs. The problem formulation (1) is closely related to the average-reward objective in semi-Markov decision processes (Puterman, 1994), where sojourn times are state-dependent random variables. Our policy explicitly parameterises the sojourn time distribution and optimises it online, but without requiring a full semi-MDP transition model. Hyperbolic geometry as uncertainty proxy. The curvature signal κ acts as a calibrated uncertainty estimate without requiring a probabilistic model. Unlike entropy-based or ensemble disagreement methods, it is computed geometrically and scales (n2)O(n^2) in the number of futures, which is manageable for small n (we use n=4n=4). It is also differentiable w.r.t. the embedding, opening a path to end-to-end learning of the embedding jointly with the pacing policy. Broader applicability. Autonomous agents that must allocate attention, digital assistants, robotic planners, multi-agent systems with communication costs, face the same decision: when to act next. ATCPG is lightweight (one weight vector, one phase scalar, one curvature call) and can be dropped into any agent loop that already exposes a world model or planning module. 9 Conclusion In continuous autonomous operation, an agent must decide not only what action to take, but also when to act next. While recent LLM-based and multi-agent systems research has primarily emphasised orchestration and routing, the complementary question of self-directed temporal pacing is less often treated as a learnable decision variable. This work introduces ATCPG, a lightweight, learnable pacing layer that treats the inter-tick interval as a first-class control variable, enabling agents to regulate their own cognitive update frequency. ATCPG combines a simple online linear pacing policy with an interval-aware shaping reward that addresses a directional failure mode in temporal credit assignment that can arise under naïve outcome-based rewards. We show that, in pacing settings, such rewards can be misaligned with the desired “wait-when-needed” behaviour, systematically eroding the incentive to slow down under load. By explicitly pricing the chosen interval, ATCPG mitigates this pathology and yields substantial standalone efficiency gains in ablation. A central component of the framework is a predictive hyperbolic-spread signal κ, computed by embedding sampled candidate futures in the Poincaré ball. The geometry selectively amplifies directional disagreement near the boundary while compressing confident and near-origin states, yielding a practical proxy for prospective uncertainty. Both theoretical analysis and empirical results support the use of this geometric signal as a timing-relevant indicator of epistemic conflict. We further introduce ATCPG-ST, a joint spatio-temporal extension that augments the policy state with embedded position trajectories. This spatial signal provides additional structure for pacing decisions, yielding consistent improvements in efficiency without modifying the learning rule. Across ablations, we observe a monotonic improvement chain, indicating that learning, geometric uncertainty, and spatial context contribute complementary gains. Across evaluated settings, adaptive pacing improves efficiency over fixed-interval baselines at matched success rates, with the full ATCPG-ST variant achieving the strongest gains. These effects also transfer beyond synthetic settings: in a small-scale LLM-agent experiment using a commercial API deployment, ATCPG reduces token consumption while maintaining identical task success rates, indicating that adaptive pacing can translate into practical reductions in runtime cost. ATCPG is intentionally minimal, requiring only a single weight vector, a phase variable, and a curvature computation, and can be integrated into any agent loop capable of producing candidate futures. Taken together, these results support adaptive pacing as a distinct layer of agent behaviour that complements decision-making and planning by regulating the frequency of re-evaluation. Limitations and Future Work. We note three specific boundaries to the current empirical validation. First, the real-world LLM benchmark (Section 7.9) operates at a small scale (N=15N=15 episodes) on a task where both the baseline and adaptive controllers achieve a 100%100\% success rate. While this cleanly isolates the efficiency gains (reducing token consumption by 10.2%10.2\% without degrading performance), it leaves open the question of how ATCPG behaves in highly complex, low-success-rate environments where the agent might need to actively trade temporal efficiency for increased reasoning accuracy. Second, ATCPG is designed strictly as a systems-level efficiency and orchestration layer; it does not improve the underlying reasoning capability or zero-shot accuracy of the LLM itself. 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[2025] V. Erofeeva, O. Granichin, V. Pankov, and Z. Volkovich. Communication-efficient decentralized clustering for dynamical multi-agent systems. PLOS ONE, 20(7):e0327396, 2025. https://doi.org/10.1371/journal.pone.0327396. Appendix A Proof of Proposition 1 (Extended) We provide a full worked example. Suppose: =[θbias,θf]=[60,30] θ=[ _bias, _f]=[60,30], ft=5f_t=5 (heavy fatigue), Δwt=−0.3 w_t=-0.3 (overload), α=0.1α=0.1. Naive outcome reward: rt=2Δwt=−0.6r_t=2 w_t=-0.6. Update: θf←30+0.1⋅(−0.6)⋅5=30−0.3=29.7 _f← 30+0.1·(-0.6)· 5=30-0.3=29.7. Interval output: Δ^t=60+29.7⋅5=208.5 t=60+29.7· 5=208.5\,s (decreased by 1.51.5\,s). Repeated across 100100 similar ticks, the fatigue weight erodes to ≈0≈ 0, and the overloaded agent ticks as fast as an idle one. Interval-aware reward: Take Δt=60 t_t=60 (base interval), Δtbase=60 t_base=60: efficiency =2⋅(−0.3/60)=−0.010, =2·(-0.3/60)=-0.010, spacing =1.5⋅0.3⋅(60/60)=+0.450, =1.5· 0.3·(60/60)=+0.450, curv brake =0(κt=0), =0 ( _t=0), rt r_t =−0.010+0.450=+0.440. =-0.010+0.450=+0.440. Update: θf←30+0.1⋅(+0.440)⋅5=30.22 _f← 30+0.1·(+0.440)· 5=30.22. The fatigue weight increases, lengthening the interval under overload as desired. Appendix B Algorithm Pseudocode Algorithm 1 gives the complete ATCPG control loop. Algorithm 1 ATCPG: Adaptive Temporal Control via Predictive Geometry 1: Require Initial weights θ, world model W, base interval Δtbase t_base 2: ϕ←Uniform(0,2π)φ (0,2π); ω←0.05ω← 0.05 3: for t=1,2,…t=1,2,… do 4: Observe: priority ptp_t, fatigue ftf_t, wellbeing wtw_t, performance ρt _t 5: κt←ComputeCurvature(,st) _t← ComputeCurvature(W,s_t) (Def. 1; Eq. (14)) 6: st←[pt,ft,Δwt,ρt,sin⁡ϕ,κt]s_t←[p_t,f_t, w_t, _t, φ, _t] 7: Δ^t←clip(⊤st,Δtmin,Δtmax) t_t ( θ s_t,\; t_ , t_ ) (Eq. (3)) 8: With prob. εeff(st) _eff(s_t): Δ^t×=Uniform(0.5,1.5) t_t ×=Uniform(0.5,1.5) (Eq. (6)) 9: Sleep Δ^t t_t seconds; execute cognitive tick 10: Observe wt+1w_t+1; Δwt←wt+1−wt w_t← w_t+1-w_t 11: rt←IntervalAwareReward(Δwt,κt,Δ^t)r_t← IntervalAwareReward( w_t, _t, t_t) (Eq. (9)) 12: ←clip(+αrtst,[−100,100]7) θ ( θ+α r_ts_t,\;[-100,100]^7) (Eq. (4)) 13: ϕ←(ϕ+ω)mod2πφ←(φ+ω) 2π; ω←clip(ω+αrt⋅0.01,[0.001,0.2])ω (ω+α r_t· 0.01,\;[0.001,0.2]) 14: end for Implementation notes. All experiments were implemented in Python using standard numerical and scientific-computing libraries (e.g., NumPy) and were executed under a controlled set of random seeds to enable consistent comparisons across ablations. Due to proprietary constraints, the current implementation is not publicly released; we intend to open-source a reference implementation pending internal legal and compliance clearance. In the interim, the controller, reward, and curvature estimator are specified fully in closed form (Sections 4–6.5), and all environmental and algorithmic hyperparameters required to replicate the reported experimental protocol are provided in Section 7.

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